weyl, eigenfunction expansions and harmonic …symmetric spaces of rank 1. it is the purpose of this...

2

Weyl, eigenfunction expansions andharmonic analysis on non-compact

symmetric spacesErik van den Ban

University of Utrecht

[email protected]

1 Introduction

This text grew out of an attempt to understand a remark by Harish-Chandra in the introduction of [12]. In that paper and its sequel hedetermined the Plancherel decomposition for Riemannian symmetricspaces of the non-compact type. The associated Plancherel measureturned out to be related to the asymptotic behavior of the so-calledzonal spherical functions, which are solutions to a system of invariantdifferential eigenequations. Harish-Chandra observed: ‘this is reminis-cent of a result of Weyl on ordinary differential equations’, with referenceto Hermann Weyl’s 1910 paper, [29], on singular Sturm–Liouville oper-ators and the associated expansions in eigenfunctions.

For Riemannian symmetric spaces of rank one the mentioned systemof equations reduces to a single equation of the singular Sturm–Liouvilletype. Weyl’s result indeed relates asymptotic behavior of eigenfunctionsto the continuous spectral measure but his result is formulated in asetting that does not directly apply.

In [23], Kodaira combined Weyl’s theory with the abstract Hilbertspace theory that had been developed in the 1930’s. This resulted inan efficient derivation of a formula for the spectral measure, previouslyobtained by Titchmarsh. In the same paper Kodaira discussed a class ofexamples that turns out to be general enough to cover all Riemanniansymmetric spaces of rank 1.

It is the purpose of this text to explain the above, and to describelater developments in harmonic analysis on groups and symmetric spaceswhere Weyl’s principle has played an important role.

23

24 Eigenfunction expansions

2 Sturm–Liouville operators

A Sturm–Liouville operator is a second order ordinary differential oper-ator of the form

L = − d

dtpd

dt+ q, (2.1)

defined on an open interval ] a, b [ , where −∞ ≤ a < b ≤ +∞. Here p isassumed to be a C1-function on ] a, b [ with strictly positive real values;q is assumed to be a real valued continuous function on ] a, b [ .

The operator L is said to be regular at the boundary point a if a isfinite, p extends to a C1-function [ a, b [ → ] 0,∞ [ and q extends to acontinuous function on [a, b [ . Regularity at the second boundary pointb is defined similarly. The operator L is said to be regular if it is regularat both boundary points. In the singular case, no conditions are imposedon the behavior of the functions p and q towards the boundary pointsapart from those already mentioned.

The operator L is formally symmetric in the sense that

〈Lf , g〉[a,b] = 〈f , Lg〉[a,b]

for all compactly supported C2-functions f and g on ] a, b [ . Here wehave denoted the standard L2-inner product on [a, b] by

〈f , g〉[a,b] =∫ b

a

f(t) g(t) dt.

For arbitrary C2-functions f and g on ] a, b [ it follows by partial inte-gration that

〈Lf , g〉[x,y] − 〈f , Lg〉[x,y] = [f, g]y − [f, g]x, (2.2)

for all a < x ≤ y < b. Here the sesquilinear form [ · , · ]t on C1( ] a, b [ )(for a < t < b) is defined by

[f, g]t := p(t) [f(t) g′(t)− f ′(t) g(t)]. (2.3)

To better understand the nature of this form, let 〈 · , · 〉 denote the stan-dard Hermitian inner product on C2, and define the (anti-symmetric)sesquilinear form [ · , · ] on C2 by

[v, w] := 〈Jv , w〉, J =(

0 −11 0

). (2.4)

Define the evaluation map εt : C1( ] a, b [ ) → C2 by

εt(f) := ( f(t) , p(t)f ′(t) ). (2.5)

3 The case of a regular operator 25

Then the form (2.3) is given by [f, g]t = [ εt(f) , εt(g) ]. We now observethat for ξ a non-zero vector in R2,

〈εt(f) , ξ〉 = 0 ⇐⇒ εt(f) ∈ C · Jξ. (2.6)

Hence, if f, g are functions in C1( ] a, b [ ), then by anti-symmetry of theform [ · , · ] we see that

〈εt(f) , ξ〉 = 〈εt(g) , ξ〉 = 0 =⇒ [f, g]t = 0. (2.7)

For a complex number λ ∈ C we denote by Eλ the space of complexvalued C2-functions f on ] a, b [, satisfying the eigenequation Lf = λf.

This eigenequation is equivalent to a system of two linear first orderequations for the function ε(f) : t 7→ εt(f). It follows that for everya < c < b and every v ∈ C2 there is a unique function s(λ, · )v =sc(λ, · )v ∈ C2( ] a, b [ ) such that

s(λ, · )v ∈ Eλ, and εc(s(λ, · )v) = v. (2.8)

By uniqueness, s(λ)v depends linearly on v, and by holomorphic param-eter dependence of the system, the map λ 7→ s(λ)v is entire holomorphicfrom C to C2( ] a, b [ ).

3 The case of a regular operator

After these preliminaries, we recall the theory of eigenfunction expan-sions for a regular Sturm–Liouville operator L on [a, b]. Let ξa, ξb betwo non-zero vectors in R2. We consider the linear space C2

ξ ([a, b]) ofC2-functions f : [a, b] → C satisfying the homogeneous boundary condi-tions

〈εa(f) , ξa〉 = 0, 〈εb(f) , ξb〉 = 0. (3.1)

For all functions f and g in this space, and for t = a, b, we now have theconclusion of (2.7). In view of (2.2), this implies that L is symmetric onthe domain C2

ξ ([a, b]), i.e.,

〈Lf , g〉[a,b] = 〈f , Lg〉[a,b],

for all f, g ∈ C2ξ ([a, b]). In this setting we have the following result on

eigenfunction expansions. Let σ(L, ξ) be the set of λ ∈ C for which theintersection Eλ,ξ := Eλ ∩ C2

ξ ([a, b]) is non-trivial.


Theorem 3.1 The set σ(L, ξ) is a discrete subset of R without accu-mulation points. For each λ ∈ σ(L, ξ) the space Eλ,ξ is one dimensional.Finally,

L2([a, b]) = ⊕λ∈σ(L,ξ) Eλ,ξ (orthogonal direct sum). (3.2)

We will sketch the proof of this result; this allows us to describe whatwas known about the spectral decomposition associated with a Sturm–Liouville operator when Weyl entered the scene.

For λ ∈ C, let ϕλ be the function in Eλ determined by εa(ϕλ) = Jξa.

Then 〈εa(ϕλ) , ξa〉 = 0, hence [ϕλ, ϕλ]a = 0. The function λ 7→ ϕλ isentire holomorphic with values in C2([a, b]). We observe that Eλ,ξ 6= 0 ifand only if ϕλ belongs to Eλ,ξ, in which case Eξ,λ = Cϕλ.We thus see thatthe condition λ ∈ σ(L, ξ) is equivalent to the condition 〈εb(ϕλ) , ξb〉 = 0.

The function χ : λ 7→ 〈εb(ϕλ) , ξb〉 is holomorphic with values in C,and from (2.2) we deduce that

(λ− λ) 〈ϕλ , ϕλ〉[a,b] = [ϕλ, ϕλ]b.

In view of (2.7) we now see that the function χ does not vanish forImλ 6= 0. Its set of zeros, which equals σ(L, ξ), is therefore a discretesubset of R without accumulation points. Replacing L by a translateL + µ with −µ ∈ R \ σ(L, ξ) if necessary, we see that without loss ofgenerality we may assume that 0 /∈ σ(L, ξ). This implies that L is injec-tive on C2

ξ ([a, b]). Let g ∈ C([a, b]) and consider the equation Lf = g.

Writing this equation as a system of first order equations in terms ofε(f), using a fundamental system for the associated homogeneous equa-tion, and applying variation of the constant one finds a unique solutionf ∈ C2

ξ ([a, b]) to the equation. It is expressed in terms of g by an integraltransform G of the form

Gg(t) =∫ b

a

G(t, τ) g(τ) dτ,

with integral kernel G ∈ C([a, b] × [a, b]), called Green’s function. Theoperator G turns out to be a two-sided inverse to the operator L :C2

ξ ( ] a, b [ ) → C([a, b]).It follows from D. Hilbert’s work on integral equations, [21], that the

map (f, g) 7→ 〈f , Gg〉[a,b] may be viewed as a non-degenerate Hermi-tian form in infinite dimensions, which allows a diagonalization overan orthonormal basis ϕk of L2([a, b]), with associated non-zero diago-nal elements λk, for k ∈ N. In today’s terminology we would say thatthe operator G is symmetric and completely continuous, or compact, and

4 The singular Sturm–Liouville operator 27

Hilbert’s result has evolved into the spectral theorem for such operators.From this the result follows with σ(L, ξ) = λ−1

k | k ∈ N.

4 The singular Sturm–Liouville operator

We now turn to the more general case of a (possibly) singular operator Lon ] a, b [ . Weyl had written a thesis with Hilbert, leading to the paper[28], generalizing the theory of integral equations to ‘singular kernels.’It was a natural idea to apply this work to singular Sturm–Liouvilleoperators. At the time Weyl started his research it was understoodthat the regular cases involved discrete spectrum. On the other hand,from his work on singular integral equations it had become clear thatcontinuous spectrum had to be expected.

Also, if one considers the example with a = 0, b = ∞, and p = 1, q =0, then L = −d2/dt2 is regular at 0 and singular at ∞. Fix the boundarydatum ξ0 = (0, 1). Then one obtains the eigenfunctions cos

√λt of L,

with eigenvalue λ ≥ 0. In this case a function f ∈ C2c ([0,∞ [ ), satisfying

the boundary condition 〈ε0(f) , ξ0〉 = 0 admits the decomposition

f(t) =∫ ∞

0

a(√λ) cos(

√λt)

dλ

π√λ

involving the continuous spectral measure dλπ√

λ. Here of course, the func-

tion a is given by the cosine transform

a(√λ) =

∫ ∞

0

f(t) cos√λt dt.

Thus, no boundary condition needs to be imposed at infinity. At thetime, Weyl faced the task to unify these phenomena, where both dis-crete and continuous spectrum (in his terminology ‘Punktspektrum’ and‘Streckenspektrum’) could occur, and to clarify the role of the boundaryconditions. Finally, the question arose what could be said of the spectralmeasure.

In [29], Weyl had the important idea to construct a Green operatorfor the eigenvalue problem Lf = λf with λ a non-real eigenvalue. Hefixed boundary conditions for the Green kernel depending on a beautifulgeometric classification of the situation at the boundary points whichwe will now describe. We will essentially follow Weyl’s argument, butin order to postpone choosing bases, we prefer to use the language ofprojective space rather than refer to affine coordinates as Weyl did in[29], p. 226. The reader may consult the appendix for a quick review of


the description of circles in one dimensional complex projective space interms of Hermitian forms of signature type (1, 1).

Returning to the singular Sturm–Liouville problem, we make the fol-lowing observation about real boundary data at a point x ∈ ] a, b [ .

Lemma 4.1 Let λ ∈ C and let f ∈ Eλ \ 0. Then the followingassertions are equivalent.

(a) ∃ ξ ∈ R2 \ 0 : 〈εx(f) , ξ〉 = 0;(b) [εx(f)] ∈ P1(R);(c) [f, f ]x = 0.

Proof As [f, f ]x = [εx(f), εx(f)], these are basically assertions aboutC2, which are readily checked.

It follows that the zero set

Cλ,x := f ∈ Eλ | [f, f ]x = 0

defines a circle in the projective space P(Eλ). Indeed, let εx : P(Eλ) →P1(C) be the projective isomorphism induced by the evaluation map(2.5), then εx(Cλ,x) = P1(R).

The following important observation is made in Weyl’s paper [29],Satz 1.

Proposition 4.1 Let λ ∈ C \R. Then the circle Cλ,x in P(Eλ) dependson x ∈ ] a, b [ in a continuous and strictly monotonic fashion. Moreover,if x→ b then Cλ,x tends to either a circle or a point. A similar statementholds for x→ a.

We shall denote by Cλ,b the limit of the set Cλ,x for x → b. Thenotation Cλ,a is introduced in a similar fashion. The proof of the aboveresult is both elegant and simple.

Proof We fix a point c ∈ ] a, b [ . For x ∈ ] c, b [ we define the Hermitianinner product 〈 · , · 〉x on Eλ by 〈f , g〉x = 〈f , g〉[c,x]. It follows from(2.2) that

[f, f ]x = [f, f ]c + 2i Im (λ)〈f , f〉x,

for f ∈ Eλ and x ∈ ] c, b [ . Without loss of generality, let Im (λ) > 0.Then it follows that x 7→ −i[f, f ]x is a real valued, strictly increasingcontinuous function. All results follow from this.


In his paper [29], Weyl uses a basis f1, f2 ∈ Eλ such that εc(f2), εc(f1)is the standard basis of C2. Then [f1, f2]c = 1. In the affine chart deter-mined by f1, f2 the circle Cλ,c equals the real line. The circles Cλ,x there-fore form a decreasing family of circles which are either all contained inthe upper half plane or in the lower half plane. The form i[ · , · ]x is withrespect to the basis f1, f2 given by the Hermitian matrix Hkl = i[fk, fl]x.It follows that the center of Cλ,x is given by i[f1, f2]x/(−i[f1, f1]x), see(11.2). If Imλ > 0 then the denominator of this expression is positivefor t > c whereas the numerator has limit i for x ↓ c. It follows that inthe affine coordinate z parametrizing f2 + zf1 the circles Cλ,x lie in theupper half plane. Likewise, for Imλ < 0 all circles lie in the lower halfplane.

The limit of Cλ,x as x tends to one of the boundary points is closelyrelated to the L2-behavior of functions from Eλ at that boundary point.

Lemma 4.2 Let λ ∈ C\R, and let f ∈ Eλ \0 be such that Cf ∈ Cλ,b.

Then f ∈ L2([c, b [ ) for all c ∈ ] a, b [ .

Proof We may fix a basis f1, f2 of Eλ such that in the associated affinechart, Cλ,c corresponds to the real line. Then for every x 6= c the circleCλ,x is entirely contained in the associated affine chart. There exists asequence of points xn ∈ ] c, b [ and Fn ∈ Cλ,xn

such that xn → b andFn → F := Cf.

We agree to write fz = zf1 + f2. Then there exist unique zn ∈ C suchthat Fn = Cfzn

. Now zn converges to a point z∞ and F = Cfz∞ . Form < n we have

〈fzn, fzn

〉xm≤ 〈fzn

, fzn〉xn

= −(λ− λ)−1 [fzn, fzn

]c.

The expression on the right-hand side has a limit L for n→∞. It followsthat

〈fz∞ , fz∞〉xm ≤ L.

This is valid for any m. Taking the limit for xm → b we conclude thatfz∞ ∈ L2([c, b [ ).

Lemma 4.3 Let λ ∈ C \ R and assume that Eλ|[c,b [ ⊂ L2([c, b [ ).

(a) The Hermitian form hλ,x := i[ · , · ]x|Eλhas a limit hλ,b = i[ · , · ]λ,b,

for x→ b.

(b) The form hλ,b is Hermitian and non-degenerate of signature (1, 1).(c) The limit set Cλ,b is the circle given by [f, f ]λ,b = 0.


(d) In the space Hom(E∗λ, Eλ) the inverse h−1λ,x converges to h−1

λ,b asx→ b.

Proof (a) From (2.2) it follows by taking the limit for x→ b that

[f, g]λ,b = limx→b

[f, g]x = [f, g]c + (λ− λ)〈f , g〉[c,b]

for all f, g ∈ Eλ. This establishes the existence of the limit hλ,b. As hλ,x

is a Hermitian form for every x ∈ ] a, b [ , the limit is Hermitian as well.(b, d) Fix a basis f1, f2 of Eλ and write f(z) := z1f1+z2f2, for z ∈ C2.

For x ∈ [c, b] we define the Hermitian matrix Hx by i[f(z), f(w)]x =〈Hxz , w〉. Then Hx → Hb as x→ b. We will finish the proof by showingthat detHx is a constant function of x ∈ [c, b [ , so that detHb = detHc <

0 and moreover H−1x → H−1

b .

Write εx(f) for the linear endomorphism of C2 given by z 7→ εx(f(z)).Then

[f(z), f(w)]x = 〈J εx(f)z , εx(f)w〉 = 〈εx(f)∗J εx(f)z , w〉,

so that Hx = i εx(f)∗Jεx(f). It follows that detHx = −|det εx(f)|2. Bya straightforward calculation one sees that det εx(f) = [f1, f2]x. NowLf2 = λf2, so that from (2.2) it follows that [f1, f2]x − [f1, f2]c = 0 forall x ∈ [c, b [ . Hence det εx(f) = det εc(f) for all x ≥ c.

Finally, the proof of (c) is straightforward.

Combining Lemmas 4.2 and 4.3 we obtain the following corollary.

Corollary 4.4 Let λ ∈ C \ R. Then precisely one of the followingstatements is valid.

(a) The limit set Cλ,b is a circle. For any c ∈ ] a, b [ the space Eλ|[c,b [

is contained in L2([c, b [ ).(b) The limit set Cλ,b consists of a single point. For any c ∈ ] a, b [

the intersection of Eλ|[c,b [ with L2([c, b [ ) is one dimensional.

At a later stage in his paper, [29], Satz 5, Weyl used spectral consid-erations to conclude that if (a) holds for a particular non-real eigenvalueλ ∈ C, then Eλ|[c,b [ consists of square integrable functions for any eigen-value λ. We will return to this in the next section, see Lemma 6.2. Itfollows that the validity of (a), and hence the validity of the alternative(b), is independent of the particular choice of the non-real eigenvalue λ.

If (a) holds, the operator L is said to be of limit circle type at b(‘Grenzkreistypus’), and if (b) holds, L is said to be of limit point type


at b (‘Grenzpunkttypus’). With obvious modifications, similar resultsand terminology apply to the other boundary point, a. We note thata regular Sturm–Liouville operator is of the limit circle type at bothboundary points.

Weyl observed that for each boundary point, the type of L determineswhether boundary conditions should be imposed or not. Indeed, if L is ofthe limit point type at the boundary point, then no boundary conditionis needed there. On the other hand, if L is of the limit circle typeat a boundary point, then a boundary condition is required to ensureself-adjointness. Following Weyl, we shall now describe how boundaryconditions can be imposed in the limit circle case.

The idea is to fix a non-real eigenvalue λ ∈ C and to construct a Greenfunction for the operator L− λI. Weyl did this for the particular valueλ = i, but observed that the method works for any choice of non-realλ, see [29], text above Satz 5. In the mentioned paper Weyl considersthe case a = 0, b = ∞, and L regular at a, but the method works ingeneral. In what follows, our treatment will deviate from Weyl’s withregard to technical details. However, in spirit we will stay close to hisoriginal method.

We define D to be the space of functions f ∈ C1( ] a, b [ ) such thatf ′ is locally absolutely continuous (so that Lf is locally integrable).Moreover, we define Db to be the subspace of functions f ∈ D suchthat both f and Lf are square integrable on [c, b [ for some (hence any)c ∈ ] a, b [ . The subspace Da is defined in a similar fashion.

Given two functions f, g ∈ Db, it follows by application of (2.2) that

[f, g]b := limx→b

[f, g]x

exists. If χ ∈ Db, then we denote by Db(χ) the space of functions f ∈ Db

such that [f, χ]b = 0. We now select a non-zero function ϕb,λ ∈ Eλ suchthat the associated point Cϕb,λ ∈ P(Eλ) belongs to the limit set Cλ,b. Itis possible to characterize the function ϕb,λ by its limit behavior towardsb.

Lemma 4.5

(a) If L is of limit point type at b, then Eλ ∩ Db = Cϕb,λ.

(b) If L is of limit circle type at b, then there exists a function χb ∈ Db

such that Eλ ∩ Db(χb) = Cϕb,λ.

Proof (a) follows from Corollary 4.4. For (b), assume that L is of limit


circle type at b. Then Eλ ⊂ Db. Take χb = ϕb,λ. Then the space on theleft-hand side of the equality equals the space of f ∈ Eλ with [f, χb]c = 0.The latter space is one dimensional since [ · , · ]b is non-degenerate onEλ. On the other hand, ϕb,λ belongs to it, by Lemma 4.3, and the resultfollows.

To make the treatment as uniform as possible, we agree to always usethe dummy boundary datum χb = 0 in case L is of limit point type atb. In the limit circle case we select χb as in Lemma 4.5 (b). Then wealways have

Eλ ∩ Db(χb) = Cϕb,λ.

We follow the similar convention for a choice of boundary datum χa ∈Da, so that Da(χa) ∩ Eλ is a line representing a point of the limit setCλ,a. Moreover, we choose a non-zero eigenfunction ϕa,λ spanning thisline. Before proceeding we observe that it follows from Proposition 4.1that

Cλ,a ∩ Cλ,b = ∅.

This implies that ϕa,λ and ϕb,λ form a basis of Eλ. The above choiceshaving been made, we put

Dχ = Da(χa) ∩ Db(χb).

Then Dχ is a subspace of L2( ] a, b [ ). It contains C2c ( ] a, b [ ), hence is

dense. Moreover, it follows from the above that

Dχ ∩ Eλ = 0.

Still under the assumption that λ ∈ C \ R, we now consider the dif-ferential equation

(L− λ)f = g (4.1)

where g is a given square integrable function on ] a, b [ . The equationmay be rewritten as a first order equation for the C2-valued functionε(f). The matrix with columns ε(ϕa,λ) and ε(ϕb,λ) is a fundamentalmatrix for this system. By variation of the constant one finds a functionf ∈ D, satisfying (4.1). If g has compact support then f can be uniquelyfixed by imposing the boundary conditions

limx→a

[f, χa]x = 0, limx→b

[f, χb]x = 0.


This function is expressed in terms of g by means of an integral operator,

f(t) = Gλg(t) :=∫ b

a

Gλ(t, τ) g(τ) dτ, (4.2)

whose integral kernel, called the Green function, is given by

Gλ(t, τ) = w(λ)−1 ϕa,λ(t)ϕb,λ(τ), (t ≤ τ), (4.3)

and Gλ(t, τ) = Gλ(τ, t) for t ≥ τ. Here w(λ) is the Wronskian, definedby

w(λ) = [ϕb,λ, ϕa,λ]c,

for a fixed c ∈ ] a, b [ ; note that the expression on the right-hand sideis independent of c, by (2.2). It is an easy matter to show that Gλ iswell defined on L2( ] a, b [ ), with values in D. Moreover, (L−λI)Gλ = I.

Finally, Gλ maps functions with compact support into Dχ.

At this point Weyl essentially proves the following result. He special-izes to λ = i and splits Gλ into real and imaginary part, but the crucialidea is to approximate the Green kernel by Green kernels associated toa regular Sturm–Liouville problem on smaller compact intervals, wherethe spectral decomposition of Theorem 3.1 is applied.

Theorem 4.2 The Green operator Gλ is a bounded linear endomor-phism of L2( ] a, b [ ), with operator norm at most |Imλ|−1.

Proof For z ∈ C we consider the eigenfunction ϕzb = ϕb,λ + zϕa,λ.

As Cϕb,λ is contained in the limit set Cλ,b, there exists a sequence ofpoints bn ∈ ] a, b [ and zn ∈ C such that bn b, zn → 0 and ϕn

b := ϕzn

b

represents a point of the circle Cλ,bn. Similarly, there is a sequence of

points an ∈ ] a, b [ , wn ∈ C such that an a, wn → 0 and ϕna :=

ϕa,λ + wnϕb,λ represents a point of Cλ,an . Define Gnλ as in (4.3), but

with ϕa,λ and ϕb,λ replaced by ϕna and ϕn

b respectively. Then it isreadily seen that Gn

λ → Gλ, locally uniformly on ] a, b [× ] a, b [ . Applythe spectral decomposition associated with the regular Sturm–Liouvilleproblem for L on [an, bn] with boundary data ϕn

a and ϕnb . Then the

operator Gnλ : L2([an, bn]) → L2([an, bn]) satisfies (L−λ) Gn

λ = I, hencediagonalizes with eigenvalues (ν − λ)−1, ν ∈ R. All of these eigenvalueshave length at most |Imλ|−1, so that ‖Gn

λ‖ ≤ |Imλ|−1. It now followsby taking limits that for all f, g ∈ Cc( ] a, b [ ),

|〈f , Gλg〉| = limn→∞

|〈f , Gnλg〉‖ ≤ |Imλ|.


This implies the result.

Corollary 4.6 The operator Gλ is a bounded linear endomorphism ofthe space L2( ] a, b [ ) with image equal to Dχ. Moreover, Gλ is a two-sidedinverse to the operator L− λI : Dχ → L2( ] a, b [ ).

Proof It is easy to check that Gλ is continuous as a map L2( ] a, b [ ) →C1( ] a, b [ ). Using (2.2) and Theorem 4.2 it is then easy to check thatg 7→ [Gλg, χb]b is continuous on L2( ] a, b [ ). As this functional vanisheson functions with compact support, it follows that Gλ maps L2( ] a, b [ )into Db(χb). By a similar argument at the other boundary point weconclude that Gλ maps into Dχ.

We observed already that (L − λI)Gλ = I on L2( ] a, b [ ). It followsthat (L−λI) [Gλ(L−λI)−I] = 0 on Dχ. As Gλ maps into Dχ, on whichL− λI is injective, it follows that Gλ(L− λI)− I on Dχ. All assertionsfollow.

Looking at Weyl’s result from a modern perspective, it is now possibleto show that the densely defined operator L with domain Dχ is self-adjoint. To prepare for this, we need a better understanding of theboundary conditions.

In what follows we will assume that L is of limit circle type at b, so thatEλ ⊂ Db. For x ∈ ] a, b [ , the map εx : Eλ → C2 is a linear isomorphism.We define the map βλ,x : Db → Eλ by εx βx(f) = εx(f), for f ∈ Db.

Then βλ,x may be viewed as a projection onto Eλ. For f, g ∈ Db,

[βλ,x(f), βλ,x(g)]x = [f, g]x. (4.4)

This implies that

[βλ,x(f), · ]x = [f, · ]x on Eλ. (4.5)

As the form [ · , · ]b is non-degenerate on Eλ, we may define a linear mapβλ,b : Db → Eλ by (4.5) with x = b. Then again βλ,b = I on Eλ, so thatβλ,b may be viewed as a projection onto Eλ.

Lemma 4.7 Let f ∈ Db. Then βλ,x(f) → βλ,b(f) in Eλ, as x→ b.

Proof Let γx denote the sesquilinear form [ · , · ]x on Eλ, for a < x ≤ b.

Then for x→ b, we have the limit behavior γx → γb in Hom(Eλ, E∗λ) and

γ−1x → γ−1

b in the space Hom(E∗λ, Eλ), see Lemma 4.3.


From (4.5) we deduce that γx(βλ,x(f)) = [f, · ]x, for all f ∈ Db. Itfollows that γx(βλ,x(f)) → [f, · ]b = γb(βλ,b)(f), hence

βλ,x(f) = γ−1x γxβλ,x(f) → βλ,b(f),

for x→ b.

Corollary 4.8 For all f, g ∈ Db we have [f, g]b = [βλ,b(f), βλ,b(g)]b.

Proof This follows from (4.4) by passing to the limit for x→ b.

The following immediate corollary clarifies the nature of the boundarydatum χb.

Corollary 4.9 Let χb ∈ Db. Then Db(χb) depends on χb through itsimage βλ,b(χb) in Eλ.

It follows that in the present setting (L of limit circle type at b), theequality of Lemma 4.5 (b) is equivalent to Cβλ,b(χb) = Cϕb,λ. In otherwords, let Cλ,b denote the preimage in Eλ \ 0 of the limit circle Cλ,b.

Then functions from β−1λ,b(Cλ,b) provide appropriate boundary data at b.

The following result is needed to determine the adjoint of the Greenoperator Gλ. As Eλ ⊂ Db it follows that Eλ = Eλ is contained in Db aswell. We put ϕb,λ := ϕb,λ; then the above definitions are valid with λ

instead of λ. It is immediate from the definitions that

βλ,b(f) = βλ,b(f), (f ∈ Db). (4.6)

Lemma 4.10

(a) βλ,b βλ,b = βλ,b;(b) βλ,b(ϕb,λ) = c ϕb,λ, with c a non-zero complex scalar;(c) βλ,b(χb) = c ϕb,λ;(d) Db(χb) = Db(χb).

Proof (a) We check that βλ,x = βλ,xβλ,x by applying εx on the left.Now use Lemma 4.7.

(b) Let ψ be an eigenfunction in Eλ. Then [ϕλ,b, ψ]x is constant as afunction of x, by (2.2). Hence, [βλ,b(ϕλ,b), ψ]b = [ϕλ,b, ψ]x. It follows thatβλ,b(ϕλ,b) is non-zero, whereas [βλ,b(ϕλ,b), ϕλ,b]b = [ϕλ,b, ϕλ,x]x = 0. Theassertion follows.

(c) Applying (4.6), (a) and (b) we obtain: βλ,b(χb) = βλ,b(χb) =


βλ,bβλ,b(χb) = βλ(ϕb,λ) = cϕb,λ. Finally, (d) is an immediate conse-quence of (c).

We now return to the situation of a general singular Sturm–Liouvilleoperator L.

Theorem 4.3 The operator L with domain Dχ is self-adjoint.

Proof From Lemma 4.10 it follows that Dχ = Dχ. The adjoint G∗ of theoperator G = Gλ has integral kernel G∗λ(t, τ) := Gλ(τ, t). This is preciselythe Green kernel associated with the eigenvalue λ and the boundary dataχa, χb. As its image Dχ equals Dχ, it follows that G∗ is the two-sidedinverse of the bijection L − λI : Dχ → L2( ] a, b [ ). These facts implythat the adjoint L∗ equals L.

At this point one can prove the following generalization of Theorem3.1, due to Weyl, [29], Satz 4. Let σ(L, χ) be the set of λ ∈ C for whichEλ ∩ Dχ 6= 0.

Theorem 4.4 (Weyl 1910) Let L be of limit circle type at both endpoints. Then σ(L, χ) is a discrete subset of R, without accumulationpoints. Moreover, L2( ] a, b [ ) is the orthogonal direct sum of the spacesEλ ∩ Dχ.

Proof Weyl proved this by using the Green operator G correspondingto the eigenvalue i. Let G2 be the imaginary part of its kernel. Then G2

is real valued, symmetric and square integrable, hence admits a diago-nalization. In today’s terminology, the associated integral operator G2,

which equals (2i)−1[G − G∗], is self-adjoint and Hilbert-Schmidt, hencecompact. All its eigenspaces are finite dimensional, and contained in Dχ,

since Dχ = Dχ. Moreover, each of them is invariant under the symmetricoperator L.

The regular case may be viewed as a special case of the above. Indeed,if ξa, ξb are the boundary data of Theorem 3.1, let µ ∈ C\R be arbitrary,and for x = a, b, let χx be the constant function with value Jξx. ThenEλ,ξ = Eλ ∩ Dχ, for all λ ∈ C.

5 Weyl’s spectral theorem

Using Green’s function Gλ for non-real λ and his earlier work on singularintegral equations, [28], Weyl was able to establish the existence of a

5 Weyl’s spectral theorem 37

spectral decomposition of L2( ] a, b [ ) in terms of eigenfunctions of L withreal eigenvalue. In [29] he considers the case a = 0, b = ∞, and assumesthat L is regular (hence of circle limit type) at 0. Let ξa ∈ R2 \ 0be a boundary datum at a, and fix a unit vector η ∈ R2 perpendicularto ξ. Let χ0 be the constant function with value η and let χ∞ be aboundary datum at ∞ (if L is of limit point type at ∞, we take thedummy boundary datum χ∞ = 0). For each λ ∈ C let ϕλ be the uniqueeigenfunction of L with eigenvalue λ and ϕλ(a) = η. Then accordingto Weyl, [29], Satz 5,7, there exists a right-continuous monotonicallyincreasing function ρ such that each function f ∈ C2( ] 0,∞ [ ) ∩ Dχ

admits a decomposition of the form

f(x) =∫

Rϕλ(x) dF (λ) (5.1)

with uniformly and absolutely converging integral; here dF (λ) is a reg-ular Borel measure, defined by

dF (∆) =∫ ∞

0

f(t)∫

∆

ϕλ(t) dρ(λ) dt. (5.2)

In the above, dρ denotes the regular Borel measure determined by theformula dρ( ]µ, ν]) = ρ(ν)− ρ(µ), for all µ < ν.

Actually, Weyl’s original formulation was different and involved a dis-crete and a continuous part. His formulation follows from the one aboveby the observation that ρ admits a unique decomposition ρ = ρd + ρc

with ρc a continuous monotonically increasing function with ρc(0) = 0,and with ρd a right-continuous monotonically increasing function whichis constant on each interval where it is continuous.

In case L is of the limit circle type at infinity, the decomposition isdiscrete by Theorem 4.4, so that ρc = 0, so that the above gives rise toa discrete decomposition. In case L is of the limit point type at ∞, thedecomposition is of mixed discrete and continuous type.

It has now become customary to write

dF (λ) = Ff(λ) dρ(λ), Ff(λ) =∫ ∞

0

f(t)ϕλ(t) dt, (5.3)

with the interpretation that the integral converges as an integral withvalues in L2(R, dρ).

We will call ρ the spectral function associated with the operator L,the boundary data χ0, χ∞, and the choice of eigenfunctions ϕλ. In [29],Weyl also addressed the natural problem to determine its continuouspart ρc.


Theorem 5.1 (Weyl 1910) Assume L is a Sturm–Liouville operatorof the form (2.1) on [0,∞ [ , regular at 0. Assume moreover that thecoefficients p and q satisfy the conditions

(a) limt→∞ t|p(t)− 1| = 0, limt→∞ t q(t) = 0,(b)

∫∞0

t|p(t)− 1| dt <∞,∫∞0t|q(t)| <∞.

Then L is of the limit point type at ∞. Let ξa, η, χa and ϕλ be defined asabove and let ρ be the associated spectral function. Then the support ofdρd is finite and contained in the open negative real half line ] −∞, 0 [ .The support of dρc is contained in the closed positive real half line [0,∞ [.There exist uniquely determined continuous functions a, b : ] 0,∞ [→ Rsuch that

ϕλ(t) = a(λ) cos(t√λ) + b(λ) sin(t

√λ) + o(t). (5.4)

In terms of these coefficients, the spectral measure dρc is given by

dρc(λ) =1

a(λ)2 + b(λ)2dλ

π√λ.

Here we note that by (5.4) and the condition on f, the integral (5.3)is absolutely convergent. If p = 1 and q = 0, then of course one hasa(λ) = η1 and b(λ) = η2, and one retrieves the continuous measuredρc(λ) = (π

√λ)−1dλ.

Let c(√λ) := 1

2 (a(λ)− ib(λ)). Then

ϕλ(t) = c(λ)eit√

λ + c(λ)e−it√

λ + o(t)

and the spectral measure is given by

dρ(λ) =d√λ

2π|c(√λ)|2

(5.5)

We may view the operator L as a perturbation of the operator−d2/dt2.

At infinity the eigenfunction ϕλ behaves asymptotically as a linear com-bination of the exponential eigenfunctions for the unperturbed problem,with amplitudes of equal modulus |c(

√λ)|. The spectral measure of the

perturbed problem is obtained from the spectral measure of the un-perturbed problem by dividing through |c(

√λ)|2. As we will see later,

this principle is omnipresent in the theory of harmonic analysis of non-compact Riemannian symmetric spaces, of non-compact real semisimpleLie groups, and of their common generalization, the so-called semisimplesymmetric spaces.

6 Dependence on the eigenvalue parameter 39

6 Dependence on the eigenvalue parameter

In this section we will prove holomorphic dependence of the Green func-tion Gλ on the parameter λ. This is not obvious from the definition(4.3). Indeed, in the limit circle case at b, the particular normalizationof ϕb,λ chosen only guarantees real analytic dependence on the param-eter λ (this fairly easy result will not be needed in the sequel). In thelimit point case, only the line Cϕb,λ does not depend on the choicesmade, but the dependence of ϕb,λ on λ may be arbitrary. The followingresult suggests to look for differently normalized eigenfunctions, whichdo depend holomorphically on λ.

Lemma 6.1 The Green kernel Gλ defined by (4.2) depends on ϕa,λ andϕb,λ through their images in P(Eλ).

Proof This is caused by the division by the Wronskian w(λ) = [ϕb,λ, ϕa,λ].

The following result follows by application of the method of variationof the constant as explained in [7], Thm. 2.1, p. 225. The assertionabout holomorphic dependence is not given there, but follows by thesame method of proof.

Lemma 6.2 Let a < c < b and assume that for some λ0 ∈ C theeigenspace Eλ0 |[c,b [ is contained in L2([c, b [ ). Then for each eigenvalueλ ∈ C the associated eigenspace Eλ|[c,b [ is contained in L2([c, b [ ).

Moreover, for each c ∈ ] a, b [ and all v ∈ C2, the function λ 7→sc(λ, · )v|[c,b [ (see (2.8)) is entire holomorphic as a function with valuesin L2([c, b [ ).

In the following, we assume that L is of limit circle type at b. Fromthe text below (4.5) we recall the definition of the map βλ,b : Db → Eλ,

for every λ ∈ C \ R.

Lemma 6.3 Let L be of the limit circle type at b. Then for all µ, λ ∈C \ R,

(a) βµ,b βλ,b = βµ,b;(b) the restriction βµ,b|Eλ

is a linear isomorphism onto Eµ;(c) the restriction βµ,b|Eλ

induces a projective isomorphism P(Eλ) →P(Eµ), mapping the limit circle Cλ,b onto the limit circle Cµ,b.


Proof Assertion (a) is proved in the same fashion as assertion (a) ofLemma 4.10. Since [ · , · ]b is non-degenerate on both Eλ and Eµ, assertion(b) follows by application of Corollary 4.8. Finally, (c) follows from theidentity of Corollary 4.8, in view of Lemma 4.3.

The following result suggests the modification of the eigenfunctions in(4.2) that we are looking for.

Lemma 6.4 Let L be of the limit circle type at b. Let χb ∈ Db andassume that for some µ ∈ C \ R the function βµ,b(χb) is non-zero andrepresents a point of the limit circle Cµ,b. Then

(a) for each λ ∈ C\R the function βλ,b(χ) is a non-zero eigenfunctionin Eλ which represents a point of the limit circle Cλ,b;

(b) for each c ∈ ] a, b [ , the map λ 7→ [εc(βλ,b(f))] is holomorphicfrom C \ R to P1(C) \ P1(R).

Proof By the first assertion of Lemma 6.3, βλ,b(χ) = βλ,bβµ,b(χ) =βλ,b(ϕb,µ). Assertion (a) follows by application of the remaining asser-tions of the mentioned lemma.

We now turn to (b). We will prove the holomorphy in a neighborhoodof the fixed point λ0 ∈ C \ R. As βλ,b(χ) = βλ,b(βλ0,b(χ)) we may aswell assume that χ ∈ Eλ0 and that [χ] ∈ Cλ0,b. Select a sequence xn

in ] a, b [ converging to b, and for each n a point pn ∈ Cλ0,xnsuch that

pn → [χ]. There exist χn ∈ Eλ0 such that [χn] = pn and χn → χ inEλ0 . We define ϕλ,n ∈ Eλ by εxn

ϕn,λ = εxnχn. Then in the notation of

(4.5), ϕn(λ) equals βλ,xn(χn) and represents a point of Cλ,xn . For eachfixed λ the sequence βλ,xn

|Eλ0in Hom(Eλ0 , Eλ) has limit βλ,b. Hence

ϕn(λ) → βλ,b(χ), pointwise in λ.

Let c ∈ ] a, b [ . Passing to a subsequence we may assume that xn > c

for all n ≥ 1. The map εc : P(Eλ) → P1(C) maps the circle Cλ,c ontoP1(R). Let Ω be a connected open neighborhood of λ0. Then it followsby application of Proposition 4.1 that all circles εc(Cλ,xn

) are containedin one particular connected component U of P1(C)\P1(R). This impliesthat ψn(λ) := εcβxn,λ(f) ∈ U for every λ ∈ Ω. By using an affinechart containing the compact closure of U we see that the sequence ψn

has a subsequence converging locally uniformly to a holomorphic limitfunction ψ : Ω → U. By pointwise convergence, ψ(λ) = [εcβλ,b(χ)], and(b) follows.

7 A paper of Kodaira 41

Corollary 6.5 Let L be of the limit circle type at b and let χb ∈ Db be asin the above lemma. There exists a family of functions ϕλ,b ∈ C2( ] a, b [ )depending holomorphically on λ ∈ C \ R such that for each λ ∈ C \ R

(a) ϕb,λ ∈ Eλ \ 0;(b) ϕb,λ represents the point [βλ,b(χb)] of the limit circle Cλ,b.

The following analogous result in the limit point case can be provedusing a similar method, see [7], Thm. 2.3, p. 229, for details.

Lemma 6.6 Let L be of the limit point type at b. Then there exists afamily of functions ϕb,λ ∈ C2( ] a, b [ ), depending holomorphically on theparameter λ ∈ C \ R, such that for each λ ∈ C \ R,

(a) ϕb,λ ∈ Eλ \ 0;(b) the function ϕb,λ represents the limit point in P(Eλ).

Let L be arbitrary again. We fix boundary data χa, χa as indicatedin the previous section, so that L : Dχ → L2( ] a, b [ ) is self-adjoint.Accordingly, we fix holomorphic families of eigenfunctions ϕa,λ, ϕb,λ ∈Eλ in the manner indicated in Corollary 6.5 and Lemma 6.6.

Finally, we define the Green function Gλ by means of the formula(4.3). The functions ϕa,λ, ϕb,λ used here are renormalizations of thoseused in Section 4. By Lemma 6.1 this does not affect the definition ofthe Green kernel.

Corollary 6.7 The Green kernel Gλ ∈ C( ] a, b [× ] a, b [ ) depends holo-morphically on the parameter λ ∈ C \ R.

This result of course realizes the resolvent (L−λI)−1 of the self-adjointoperator L with domain Dχ explicitly as an integral operator with kerneldepending holomorphically on λ.

7 A paper of Kodaira

For the general singular Sturm–Liouville problem, there exists a spec-tral decomposition similar to (5.1), but with a spectral matrix insteadof the spectral function ρ. Weyl observed this in [30]. The spectral ma-trix was later determined by E.C. Titchmarsh who used involved directcomputations using the calculus of residues, see [27].

Independently, K. Kodaira [23] rediscovered the result by a very el-egant method, combining Weyl’s construction of the Green functionwith the general spectral theory for self-adjoint unbounded operators


on Hilbert space, as developed in the 1930’s by J. von Neumann and M.Stone. Weyl was very content with this work of Kodaira, as becomesclear from the following quote from the Gibbs lecture delivered in 1948,[31], p. 124: ‘The formula (7.5) was rediscovered by Kunihiko Kodaira(who of course had been cut off from our Western mathematical liter-ature since the end of 1941); his construction of ρ and his proofs for(7.5) and the expansion formula [...], still unpublished, seem to clinchthe issue. It is remarkable that forty years had to pass before such athoroughly satisfactory direct treatment emerged; the fact is a reflectionon the degree to which mathematicians during this period got absorbedin abstract generalizations and lost sight of their task of finishing upsome of the more concrete problems of undeniable importance.’

We will now describe the spectral decomposition essentially as pre-sented by Kodaira [23]. Fix boundary data χa and χb as in Theorem4.3. We use the notation H := L2( ] a, b [ ). Then the operator L withdomain Dχ is a self-adjoint operator in the Hilbert space H; it thereforehas a spectral resolution dE.

To obtain a suitable parametrization of the space of eigenfunctionsfor L, fix c ∈ ] a, b [ and recall that the map εc : f 7→ (f(c), p(c)f ′(c))is a linear isomorphism from Eλ onto C2, for each λ ∈ C. We definethe function s(λ) = sc(λ, · ) : ] a, b [→ Hom(C2,C) as in (2.6). Thenλ 7→ s(λ) may be viewed as an entire holomorphic map with values inC2( ] a, b [ )⊗Hom(C2,C)). Moreover, for each λ ∈ C the map v 7→ s(λ)vis a linear isomorphism from C2 onto Eλ.

For f ∈ Cc( ] a, b [ ) and λ ∈ R we define the Fourier transform

Ff(λ) =∫ b

a

s(λ, x)∗f(x) dx, (7.1)

where s(λ, x)∗ ∈ Hom(C,C2) is the adjoint of s(λ, x) with respect to thestandard Hermitian inner products on C2 and C.

By a spectral matrix we shall mean a function P : R → End(C2) withthe following properties

(a) P (x)∗ = P (x), i.e., P (x) is Hermitian with respect to the stan-dard inner product, for all x ∈ R;

(b) P is continuous from the right;(c) P (0) = 0 and P (y)− P (x) is positive semi-definite for all x ≤ y.

Associated with a spectral matrix as above there is a unique regularBorel measure dP on R, with values in the space of positive semi-definiteHermitian endomorphisms of C2, such that dP ( ]µ, ν]) = P (ν) − P (µ)

7 A paper of Kodaira 43

for all µ ≤ ν. Conversely, a measure with these properties comes from aunique spectral matrix P. Given a spectral matrix P, we define M2 =M2,P to be the space of Borel measurable functions ϕ : R → C2 with

〈ϕ , ϕ〉P :=∫

R〈ϕ(ν) , dP (ν)ϕ(ν)〉 <∞.

Moreover, we define H = HP to be the Hilbert space completion of thequotient M2/M⊥

2 .

Let Tλ := εc Gλ. Then Tλ : H → C2 is a continuous linear map. Wedenote its adjoint by T ∗λ . Kodaira uses the elements γ1(λ), γ2(λ) of Hdetermined by prj Tλ = 〈 · , γj(λ)〉.

Theorem 7.1 (Kodaira 1949) The spectral function P determined by

dP (ν) = |ν − λ|2 Tλ dE(ν) T ∗λ (7.2)

is independent of λ ∈ C \ R. Moreover, it has the following properties.

(a) The Fourier transform extends to an isometry from the Hilbertspace H = L2( ] a, b [ ) onto the Hilbert space H = HP .

(b) The spectral resolution dE(ν) of the self-adjoint operator L withdomain Dχ is given by

F dE(S) = 1S F ,

for every Borel measurable set S ⊂ R; here 1S denotes the mapinduced by multiplication with the characteristic function of S.

For the proof of Theorem 7.1, which involves ideas of Weyl [29], we re-fer the reader to Kodaira’s paper [23]. In addition to the above, Kodairaproves more precise statements about the nature of the convergence ofthe integrals in the associated inversion formula.

After having introduced the spectral matrix, Kodaira gives an inge-nious short proof of an expression for the spectral matrix which had beenfound earlier by Titchmarsh. We observe that the C2-valued functionsFa(λ) = εc(ϕa,λ) and Fb(λ) = εc(ϕb,λ) are holomorphic functions ofλ ∈ C \R. The matrix F (λ) with columns Fa(λ) and Fb(λ) is invertiblefor λ ∈ C \ R. By the above definitions,

ϕa,λ(t) = s(λ, t)Fa(λ), ϕb,λ(t) = s(λ, t)Fb(λ). (7.3)

We now define the 2×2 matrix M(λ), the so-called characteristic matrix,


by M(λ) = −(detF )−1FaFTb , i.e.,

M(λ) = −detF (λ)−1

(Fa1Fb1 Fa1Fb2

Fa2Fb1 Fa2Fb2

)λ

(7.4)

Actually, Kodaira uses the symmetric matrix M(λ)− 12J, which has the

same imaginary part. The matrix M(λ) depends holomorphically on theparameter λ ∈ C \ R.

Theorem 7.2 (Titchmarsh, Kodaira) The spectral matrix P is givenby the following limit:

P (ν) = limδ↓0

limε↓0

1π

∫[δ,ν+δ]+iε

ImM(λ) dλ. (7.5)

Proof Multiplying both sides of (7.2) with |ν−λ|−2 and integrating overR, we find that ∫

R|ν − λ|−2 dP (ν) = TλT

∗λ .

By a straightforward, but somewhat tedious calculation, using (2.2) and[ϕa,λ, ϕa,λ]a = [ϕb,λ, ϕb,λ]b = 0, it follows that

Imλ TλT∗λ =

12i

1|[Fb, F a]|2

( [Fa, Fa]FbFT

b − [Fb, Fb]FaFT

a ).

This in turn implies that Imλ · TλT∗λ = ImM(λ). Hence,∫

R|ν − λ|−2Imλ dP (ν) = ImM(λ).

From this (7.5) follows by a straightforward argument.

After this, Kodaira shows that the above result can be extended to amore general basis of eigenfunctions. A fundamental system for L is alinear map s(λ) : C2 → Eλ, depending entire holomorphically on λ ∈ Cas a C2( ] a, b [ )-valued function, such that the following conditions arefulfilled for all λ ∈ C :

(a) s(λ)v = s(λ)(v), (v ∈ C2);(b) det(εx s(λ)) = 1, (x ∈ ] a, b [ ).

Put sj(λ) = s(λ)ej , then condition (b) means precisely that the Wron-skian [s1(λ), s2(λ)]x equals 1. Write ψ(λ) = εc s(λ) ∈ End(C2). Thenit follows that ψ entire holomorphic, and that detψ(λ) = 1. Moreover,

s(λ, x) = sc(λ, x)ψ(λ).

8 A special equation 45

We may define the Fourier transform associated with s by the identity(7.1). The associated spectral function P is expressed in terms of thespectral matrix Pc for sc by the equation

dP (λ) = ψ(λ)−1 dPc(λ) ψ(λ)∗−1.

We define the matrix F for s by the identity (7.3). Then the associatedmatrix M, defined by (7.4) is given by

M(λ) = ψ(λ)−1 Mc(λ) ψ(λ)T−1.

Kodaira shows that with these definitions, the identity (7.5) is still valid.

8 A special equation

In the second half of the paper [23], Kodaira applies the above resultsto the time independent one dimensional Schrodinger operator

L = − d2

dt2+m(m+ 1)t−2 + V (t),

with m ≥ −12 and tV (t) a real valued real analytic function on an open

neighborhood of [0,∞ [ , such that

tV (t) = O(t−ε), for t→∞,

with ε > 0. Actually, Kodaira considers a more general problem withweaker requirements both at infinity and zero, but we shall not needthis. It is in fact not clear that his condition on the behavior of Vat 0 is strong enough for the subsequent argument to be valid, as waspointed out by [22], p. 206. Kodaira’s argumentation, which we shall nowpresent, is valid under the hypotheses stated above, as they imply thatthe eigenequation Lf = λf has a regular singularity at zero. Becauseof this, the asymptotic behavior of the eigenfunctions towards zero iscompletely understood. Indeed, the associated indicial equation hassolutions m + 1 and −m, where m + 1 ≥ −m. Let c0 be any non-zeroreal constant. Then there exists a unique eigenfunction s1(λ) ∈ Eλ suchthat

s1(λ, t) = c0 tm+1ϕ(λ, t)

with ϕ(λ, · ) real analytic in an open neighborhood of 0 and ϕ(λ, 0) = 1.It can be shown that ϕ(λ, t) is entire holomorphic in λ and real valued forreal λ. Kodaira claims that there exists a second eigenfunction s2(λ) ∈Eλ, depending holomorphically on λ, such that s1, s2 form a fundamental


system fulfilling the requirements (a) and (b) stated below Theorem 7.2.Using the theory of second order differential equations with a regularsingularity this can indeed be proved along the following lines.

If k := (m + 1) − (−m) = 2m + 1 is strictly positive, there exists asecond eigenfunction s2(λ) ∈ Eλ with

s2(λ, t) = −c−10 (2m+ 1)−1 t−m +O(t−m+ε), (t→ 0).

If k is not an integer, this eigenfunction is unique. If k is an integer, thens2(λ) has a series expansion in terms of t−m+r and tm+1+s log t, (r, s ∈N), and is uniquely determined by the requirement that the coefficientof t−m+k = tm+1 is zero. Finally, if k = 0, i.e., m = − 1

2 , then thereexists a unique second eigenfunction s2(λ, t) with

s2(λ, t) = c−10 t1/2 log t+O(t1/2+ε), (t→ 0).

In all cases, by arguments involving monodromy for t around zero itcan be shown that s2(λ, t) is entire holomorphic in λ and real valuedfor real λ. Finally, from the series expansions for these functions andtheir derivatives, it follows that the Wronskian [s1(λ), s2(λ)]t behaveslike 1 + O(tε) for t → 0. Since the Wronskian is constant, this impliesthat s1, s2 is a fundamental system.

From the asymptotic behavior of s1, s2 it is seen that at the boundarypoint 0, the operator L is of limit circle type if and only if m > 1

2 . It isof limit point type if − 1

2 ≤ m ≤ 12 . In the first case we fix the boundary

datum χ0 = s1(0, · ) at 0 and in the second case we fix the (dummy)boundary datum χ0 = 0. In all cases s1 is square integrable on ] 0, 1], sothat ϕ0λ = s1(λ) and F0(λ) = (1, 0)T, in the notation of (7.3).

We now turn to the asymptotic behavior at ∞. Kodaira first showsthat for every ν with Im ν ≥ 0, ν 6= 0, there is a unique solution Φν tothe equation Lf = ν2f such that

Φν(t) ∼ eiνt, (t→∞),

the asymptotics being preserved if the expressions on both sides aredifferentiated once with respect to t. Moreover, both Φν(t) and Φ′

ν(t)are continuous in (t, ν) and holomorphic in ν for Im ν > 0.

For Im ν < 0 the function Ψν = Φν belongs to Eν2 and Ψν(t) ∼ e−iνt

for t→∞. This shows that Ψν is not square integrable towards infinity,so that L is of limit point type at infinity. We may therefore take

ϕ∞λ = Φν , (Imλ > 0, Im ν > 0, ν2 = λ).

It follows from the above that Φν(t) = a(ν) s1(ν2, t)+b(ν) s2(ν2, t), with

8 A special equation 47

a, b continuous on Im ν ≥ 0, ν 6= 0, and holomorphic on Im ν > 0. Wenote that F∞(λ) = (a(ν), b(ν))T. Using the similar expression for Φ−ν

it follows that

a(ν) = a(−ν), b(ν) = b(−ν). (8.1)

If ν is real and non-zero, then Φν and Φ−ν form a basis of Eν2 andfrom the asymptotic behavior of the (constant) Wronskian [Φν ,Φ−ν ]tone reads off that

b(ν) a(−ν)− a(ν) b(−ν) = 2iν, (ν ∈ R \ 0). (8.2)

From (8.1) and (8.2) it follows that

Im a(ν)b(ν) = −ν, (ν ∈ R \ 0). (8.3)

In particular, a and b do not vanish anywhere on R \ 0.We can now determine the spectral matrix for this problem. Indeed,

for Imλ > 0 and Im ν > 0, ν2 = λ,

F (λ) =(

1 a(ν)0 b(ν)

),

so that

ImM(λ) = −Im(a(ν)b(ν)−1 1

0 0

)=

(−Im a(ν)

b(ν) 00 0

)by (7.4). From this we conclude that the spectral matrix P (λ) has zeroentries except for the one in the upper left corner, which we denote byρ(λ). The second component of Ff now plays no role in the Plancherelformula. Indeed, define

F1f(λ) =∫ ∞

0

f(t) s1(λ, t) dt,

then we have the following.

Corollary 8.1 F1 extends to an isometry from the space L2( ] 0,∞ [ )onto L2(R, dρ). The spectral function ρ is given by

ρ(λ) = − 1π

limδ↓0

limε↓0

∫[δ,λ+δ]+iε

Ima(√µ)

b(√µ)

dµ, (8.4)

where the square root√µ with positive imaginary part should be taken.

Since a and b are holomorphic in the upper half plane, a(√µ) b(

√µ)−1

is meromorphic over the interval ] −∞, 0 [ , so that on ] −∞, 0 [ , the


measure dρ is a countable sum of point measures. Indeed, let S be the(discrete) subset of zeros for a on the positive imaginary axis i ] 0,∞ [ .Then

dρ| ]−∞,0 [ =∑σ∈S

2Resν=σν a(ν)b(ν)

· δ[σ2]

On the other hand, for λ > 0, if µ→ λ, then the integrand of (8.4) tendsto Im a(

√λ)b(

√λ)−1, with local uniformity in λ. In view of (8.3) it now

follows that

dρ(λ)| ] 0,∞ [ = − 1π

Ima(√λ)

b(√λ)

=1π

√λ dλ

|b(√λ)|2

=2π

λ d√λ

|b(√λ)|2

.

Finally, if s1(0) is not square integrable at infinity, then ρ0 := dρ(0) =0. On the other hand, if it is, then ρ0 := dρ(0) equals the squaredL2-norm of s1(0).

Finally, since s1(0, λ) is real valued for λ real, whereas Φ−ν = Φν forreal ν, there exists a real analytic function c : R \ 0 → C such that

s1(0, ν2) = c(ν)Φν + c(−ν)Φ−ν

for all ν ∈ R \ 0. This gives rise to the equationsa(ν)c(ν) + a(−ν)c(−ν) = 1b(ν)c(ν) + b(−ν)c(−ν) = 0.

Using (8.2) we now deduce that

c(ν) = −b(ν)/2iν, (ν ∈ R \ 0). (8.5)

Therefore,

dρ(λ)| ] 0,∞ [ =12π

d√λ

|c(√λ)|2

.

We thus see that the principle formulated below (5.5) still holds in thissetting.

9 Riemannian symmetric spaces

A Riemannian symmetric space is a connected Riemannian manifold Xwith the property that the local geodesic reflection at each point extendsto a global isometry of X. Up to covering, each such space allows adecomposition into a product of three types of symmetric space. Thosewith zero sectional curvature (the Euclidean spaces), those with positive

9 Riemannian symmetric spaces 49

sectional curvature (among which the Euclidean spheres) and those withnegative sectional curvature (among which the hyperbolic spaces). Itfollows from the work of E. Cartan, that the spaces of negative sectionalcurvature are precisely those given by X = G/K, where G is a connectedreal semisimple Lie group of non-compact type, with finite center, andwhere K is a maximal compact subgroup of G. The Killing form of Gnaturally induces a G-invariant Riemannian metric on G/K. The groupK is the fixed point group of a Cartan involution θ of G; this involutioninduces the geodesic reflection in the origin e = eK of X.

A typical example of a symmetric space of this type is the space X ofpositive definite symmetric n× n-matrices on which G = SL(n,R) actsby (g, h) 7→ ghgT. The stabilizer of the identity matrix equals SO(n)and the associated Cartan involution θ : G→ G is given by g 7→ (gT)−1.

The geodesic reflection in the identity matrix I is given by h 7→ h−1.

In the general setting, the derivative of the Cartan involution at theidentity element of G induces an involution θ∗ of the Lie algebra g. TheLie algebra g decomposes as a direct sum of vector spaces

g = k⊕ p,

where k and p are the +1 and −1 eigenspaces of θ∗, respectively. It canbe shown that the map

(X, k) 7→ expXk, p×K → G (9.1)

is an analytic diffeomorphism onto G. In particular, this implies thatthe exponential map induces a diffeomorphism Exp : X 7→ expXK,p → G/K. Let a be a subspace of p, maximal subject to the conditionthat it is abelian for the Lie bracket of g. Every other such subspace isK-conjugate to a. The dimension r of a is called the rank of the symmetricspace G/K.

In the example G = SL(n,R), the Lie algebra sl(n,R) consists ofall traceless n× n-matrices, and θ∗ is given by X 7→ −XT. The Cartandecomposition (9.1) is given by the decomposition of a matrix in terms ofa positive definite symmetric one times an orthogonal one. The algebraa now consists of the traceless diagonal matrices, so that the rank ofSL(n,R)/SO(n) equals n − 1. For n = 2 the space is isomorphic tothe hyperbolic upper half plane, equipped with the action of SL(2,R)through fractional linear transformations.

By a result of Harish-Chandra, the algebra D(G/K) of G-invariantlinear partial differential operators on G/K is a polynomial algebraof rank r. More precisely, let M be the centralizer of a in K, and let


W := NK(a)/M, the normalizer modulo the centralizer of a in K. Asa subgroup of GL(a), this group is the reflection group associated withthe roots of a in g. It is therefore called the Weyl group of the pair (g, a).There exists a canonical isomorphism γ from D(G/K) onto P (a∗C)W , thealgebra of W -invariants in the polynomial algebra of the complexifieddual space a∗C (equipped with the dualized Weyl group action). By aresult of C. Chevalley, the algebra P (a∗C)W is known to be polynomialof rank r.

In the example SL(n,R), the Weyl group is given by the natural ac-tion of the permutation group Sn on the space a of traceless diagonalmatrices. Here the algebra P (a∗C)W corresponds to the algebra of Sn-invariants in C[T1, . . . , Tn]/(T1 + · · ·+Tn), which is of course well knownto be a polynomial algebra of n− 1 generators of its own right. We notethat in the case of rank 1, the algebra D(G/K) consists of all polynomialsin the Laplace-Beltrami operator.

In the papers [12],[13], Harish-Chandra created a beautiful theory ofharmonic analysis for left K-invariant functions on the symmetric spaceG/K, culminating in a Plancherel formula for L2(G/K)K , the space ofleft-K-invariant functions on G/K, square integrable with respect to theRiemannian volume form. We will now give a brief outline of the mainresults.

For ν ∈ a∗C , we consider the following system of simultaneous eigenequa-tions on G/K :

Df = γ(D, iν)f, (D ∈ D(G/K)). (9.2)

For r = 1, this system is equivalent to a single eigenequation for theLaplace operator. Each eigenfunction is analytic, by ellipticity of theLaplace operator. The space of K-invariant functions satisfying (9.2) isone dimensional and spanned by the so-called elementary spherical func-tion ϕν , normalized by ϕν(eK) = 1. This function can be constructedas a matrix coefficient x 7→ 〈1K , πν(x)1K〉, with 1K a K-fixed vectorin a suitable continuous representation of G in an infinite dimensionalHilbert space, obtained by the process of induction. By Weyl invarianceof the polynomials γ(D) it follows that ϕwν = ϕν , for all w ∈W.

In terms of the elementary spherical functions one may define theso-called Fourier transform of a function f ∈ C∞c (G/K)K by

FG/Kf(ν) =∫

G/K

f(x)ϕ−ν(x)dx, (ν ∈ a∗),

with dx the G-invariant volume measure on G/K.


By analyzing the system of differential equations (9.2) it is possibleto obtain rather detailed information on the asymptotic behavior of theelementary spherical functions ϕν towards infinity. It can be shown thatthe map K/M × a → G/K, (kM,X) 7→ k expXK is surjective. Forobvious reasons, the associated decomposition

G/K = K exp a · e (9.3)

is called the polar decomposition of G/K. In it, the a-part of an el-ement is uniquely determined modulo the action of W. Let a+ be achoice of positive Weyl chamber relative to W, then it follows thatG/K = K exp a+ · e with uniquely determined a+-part. Moreover, themap K/M × a+ → G/K is an analytic diffeomorphism onto an opendense subset of G/K.

Accordingly, each elementary spherical function is completely deter-mined by its restriction to A+ := exp(a+). Moreover, the restrictedfunction ϕν |A+ satisfies the system of equations arising from (9.2) bytaking radial parts with respect to the polar decomposition (9.3). Usinga characterization of D(G/K) in terms of the universal algebra U(g),Harish-Chandra was able to analyze these radial differential equationsin great detail. This allowed him to show that, for generic ν ∈ C, thebehavior of the function ϕν towards infinity is described by

ϕν(k expX) =∑

w∈W

c(wν)e(iwν−ρ)(X)[1 +Rwν(X)], (9.4)

for k ∈ K and X ∈ a+. Here ρ ∈ a∗ is half the sum of the positive roots,counted with multiplicities, c(ν), the so-called c-function, is a certainmeromorphic function of the parameter ν ∈ a∗C , and Rν(X) is a certainanalytic function of X, depending meromorphically on the parameter ν.Moreover, the asymptotic behavior of Rν is described by

Rν(tX) = O(e−tm(X)), (X ∈ a+, t→∞), (9.5)

with m(X) a positive constant, depending on X in a locally uniformway. Each of the summands in (9.4) is an eigenfunction of the radialsystem of differential equations of its own right.

Theorem 9.1 (Harish-Chandra’s Plancherel formula). The functionc has no zeros on a∗. Moreover, let

dm(ν) :=dν

|c(ν)|2, (9.6)

with dν a suitable normalization of Lebesgue measure on a∗ (see further


down). Then dm(ν) is Weyl-group invariant and the Fourier transformFG/K extends to an isometry from L2(G/K)K onto L2(a∗, dm(ν))W .

In footnote 3), p. 242, to the introduction of his paper [12], Harish-Chandra mentions: ‘This is reminiscent of a result of Weyl [[29], p.266] on ordinary differential equations.’ It seems that Harish-Chandrawas actually very inspired by Weyl’s paper. In [5], p. 38, A. Borelwrites: ‘[...] less obviously maybe, Weyl was also of help via his workon differential equations [29], which gave Harish-Chandra a crucial hintin his quest for an explicit form of the Plancherel measure. [...] It wasthe reading of [29] which suggested to Harish-Chandra that the measureshould be the inverse of the square modulus of a function in λ describingthe asymptotic behavior of the eigenfunctions [...] and I remember wellfrom seminar lectures and conversations that he never lost sight of thatprinciple, which is confirmed by his results in the general case as well.’

It is the purpose of the rest of this section to show that for the rankone case Theorem 9.1 is in fact a rather direct consequence of Kodaira’sgeneralization of Weyl’s result, described in Section 8.

Before we proceed it should be mentioned that in [12] and [13] Theo-rem 9.1 was completely proved for spaces of rank 1. Moreover, for thesespaces the c-function was explicitly determined as a certain quotient ofGamma factors.

For spaces of arbitrary rank Theorem 9.1 was proved modulo twoconjectures. The first of these concerned the injectivity of the Fouriertransform and the second certain estimates for the c-function. The firstconjecture was proved by Harish-Chandra himself, in his work on the so-called discrete series of representations for G, [14]. The validity of thesecond conjecture followed from the work of S. Gindikin and F. Karpele-vic, [11], where a product decomposition of the c-function in terms ofrank one c-functions was established. Simpler proofs of Theorem 9.1were later found through the contributions of [20], [10], [26].

The precise normalization of the Lebesgue measure dν may be given asfollows. The polar decomposition (9.3) gives rise to an integral formula∫

G/K

f(x) dx =∫

K

∫a+f(k expX)J(X) dX dk, (9.7)

with dk normalized Haar measure on K, J a suitable Jacobian, anddX suitably normalized Lebesgue measure on a. The Jacobian J andthe measure dX are uniquely determined by the above formula and therequirement that J(tX) behaves asymptotically as et2ρ(X), for X ∈ a+


and t→∞. Let dξ denote the dual Lebesgue measure on a∗. Then

dν =1|W |

dξ

(2π)n,

with |W | the number of elements of the Weyl group.We now turn to the setting of a space of rank 1. A typical example of

such a space is the n-dimensional hyperbolic space Xn, which may berealized as the submanifold of Rn+1 given by the equation x2

1 − (x22 +

· · ·+x2n) = 1, x1 > 0. Its Riemannian metric is induced by the indefinite

standard inner product of signature (1, n) on Rn+1. As a homogeneousspace Xn ' SO(1, n)/SO(n).

More generally, as a is one dimensional, all roots in R are proportional.Let α be the simple root associated with the choice of positive chambera+. Then −α is a root as well, and possibly ±2α are roots as well. Noother multiples of α occur. We fix the unique element H ∈ a withα(H) = 1.

Via the map tH 7→ t we identify a with R; likewise, via the maptα 7→ t we identify a∗ with R ' R∗. Then a+ = ] 0,∞ [ . Rescaling theRiemannian metric if necessary we may as well assume that under theseidentifications, both dX and dξ correspond to the standard Lebesguemeasure on R.

Let m1,m2 denote the root multiplicities of α, 2α, i.e., mj is the di-mension of the eigenspace of ad(H) in g with eigenvalue j. Then withthe above identifications,

ρ =12(m1 + 2m2).

The Laplace operator ∆ satisfies γ(∆, iν) = (−‖ν‖2 − ‖ρ‖2) with ‖ · ‖the norm on a∗ dual to the norm on a induced by the Riemannian innerproduct on g/k ' p. Multiplying ∆ with a suitable negative constant,we obtain an operator L0 with γ(L0, iν) = ν2 + ρ2. Let L = L0 − ρ2,

then Lϕν = ν2ϕν .

The Jacobian J mentioned above is given by the formula

J(t) = (et − e−t)m1(e2t − e−2t)m2 .

Let L := J1/2 rad (L) J−1/2 be the conjugate of the radial part of Lby multiplication with J1/2. Put

s(ν, t) := J1/2(t)ϕν(exp tH). (9.8)

Then the system of equations (9.2) is equivalent to the single eigenequa-


tion

Ls(ν, · ) = ν2s(ν, · )

By a straightforward calculation, see [10], p. 156, it follows that

L = − d2

dt2+ q(t), q(t) =

12J−1 d

2

dt2J − 1

4J−2(

d

dtJ)2 − ρ2.

By using the Taylor series of J(t) at 0 we see that there exists a realanalytic function V on R such that

q(t) = m(m+ 1)t−2 + V (t), (t > 0),

where

m =12(m1 +m2)− 1 ≥ −1

2.

On the other hand, at infinity, J(t) equals e2tρ times a power series interms of powers of e−2t with constant term 1. From this we see thatq(t) = O(e−2t), so that V (t) = O(t−2) as t → ∞. It follows that ouroperator L satisfies all requirements of Section 8.

We now observe that ϕν(e) = 1 and J(t)1/2 ∼ 2ρtm+1 (t → 0). Letc0 := 2ρ and let s1(λ, t) be defined as in Section 8, for λ ∈ C. Then itfollows that s(ν, t) = s1(ν2, t) for all ν ∈ C. Moreover, it follows from(9.4) that the function Φν of Section 8 is given by

Φν(t) = e−tρJ(t)1/2 c(ν) eiν(1 +Rν(t)).

In particular, it depends meromorphically on the parameter ν. Fromthis it follows that the functions ν 7→ a(ν), b(ν) are meromorphic. Byanalytic continuation it now follows that the identity (8.5) extends to anidentity of meromorphic functions. From its explicitly known form as aquotient of Gamma factors, it follows that the function ν 7→ c(ν) has nozeros on i ] 0,∞ [ . Moreover, it has a zero of order 1 at 0. Using (8.5) wenow see that b has no zeros on i[0,∞ ] , so that the spectral measure dρhas no discrete part. Hence,

dρ(λ) =d√λ

2π|c(√λ)|2

∣∣∣∣∣] 0,∞ [

.

Let F1 be the Fourier transform defined in terms of s1(λ, t) = s(√λ, t),

see Section 8. Then it follows by application of (9.7) and (9.8) that

FG/Kf(ν) = F1(J1/2f exp |a+)(ν2), (ν ∈ R), (9.9)

for every f ∈ Cc(G/K).

10 Analysis on groups and symmetric spaces 55

By Corollary 8.1 the Fourier transform F1 is an isometry from thespace L2(a+, dX) onto the space L2( ] 0,∞ [ , dρ). Moreover, the mapf 7→ J1/2f exp |a+ is an isometry from L2(G/K)K onto L2(a+, dX).

Finally, since W = ±I, whereas the function ν 7→ |c(ν)|2 is even by(8.5) and (8.1), pull-back by the map ν 7→ ν2 defines an isometry

L2( ] 0,∞ [ , dρ) '−→ L2(a∗,12

dξ

2π|c(ν)|2)W . (9.10)

By (9.9) FG/K is the composition of the three mentioned isometries.The assertion of Theorem 9.1 follows.

10 Analysis on groups and symmetric spaces

After his work on the Riemannian symmetric spaces, Harish-Chandracontinued to work on a theory of harmonic analysis for real semisim-ple Lie groups in the 1960’s. His objective was to obtain an explicitPlancherel decomposition for L2(G), the space of square integrable func-tions with respect to a fixed choice of (bi-invariant) Haar measure onG.

In the case of a compact group, the Plancherel formula is described interms of representation theory and consists of the Peter–Weyl decompo-sition combined with the Schur-orthogonality relations.

In the more general case of a real semisimple Lie group, the situationis far more complicated. If G is simple and non-compact, then the non-trivial irreducible unitary representations of G are infinite dimensional.Moreover, there is a mixture of discrete and continuous spectrum.

An irreducible unitary representation is said to be of the discrete seriesif it contributes discretely to L2(G), i.e., it is embeddable as a closedinvariant subspace for the left regular representation. Equivalently, thismeans that its matrix coefficients are square integrable. An irreducibleunitary representation has a character, which is naturally defined asa conjugation invariant distribution on G. A deep theorem of Harish-Chandra in the beginning of the 1960’s asserts that in fact all suchcharacters are locally integrable. Moreover, they are analytic on theopen dense subset of regular elements.

In [14] and [15], Harish-Chandra gave a complete classification of thediscrete series. First of all, G has discrete series if and only if it has acompact Cartan subgroup. Moreover, the representations of the discreteseries are completely determined by the restriction of their charactersto this compact Cartan subgroup. Harish-Chandra achieved their clas-


sification and established a character formula on the compact Cartanwhich shows remarkable resemblance with Weyl’s character formula.

In the early 1970’s, Harish-Chandra, [16], [17], [18], completed hiswork on the Plancherel decomposition. The orthocomplement of thediscrete part of L2(G) is decomposed in terms of representations of theso-called generalized principal series. These are induced representationsof the form

πP,ξ,λ = IndGP (ξ ⊗ eiλ ⊗ 1),

where P is a (cuspidal) parabolic subgroup of G, with a so-called Lang-lands decomposition P = MPAPNP . Moreover, ξ is a discrete seriesrepresentation of MP and eiλ is a unitary character of the vectorialgroup AP . The space L2(G) splits into a finite orthogonal direct sum ofclosed subspaces L2(G)[P ], each summand corresponding to an equiva-lence class of parabolic subgroups with K-conjugate AP -part. Here Gcounts for a parabolic subgroup, and L2(G)[G] denotes the discrete partof L2(G).

Each summand L2(G)[P ] decomposes discretely into a countable or-thogonal direct sum of spaces L2(G)[P ],ξ parametrized by (equivalenceclasses of) discrete series representations of MP . Finally, each of thespaces L2(G)[P ],ξ has a continuous decomposition parametrized by λ ∈a∗P . Harish-Chandra achieved this continuous decomposition by reduc-tion to the space of functions transforming finitely under the action ofthe maximal compact subgroup K.

Let δL, δR be two irreducible representations of K and let L2(G)[P ],ξ,δ

be the part of L2(G)[P ],ξ consisting of bi-K-finite functions of left K-typeδL and right K-type δR. The decomposition of this space is describedin terms of Eisenstein integrals. These are essentially K ×K-finite ma-trix coefficients of type (δL, δR) of the induced representation involved.The Eisenstein integrals E([P ], ξ, λ, ψ) are functions on G which dependanalytically on the parameter λ ∈ a∗. In addition, they depend linearlyon a certain parameter ψ, which ranges over a certain finite dimensionalHilbert space A2(MP , ξ, δ) of functions M×K×K → C. The Eisensteinintegrals satisfy eigenequations coming from the bi-G-invariant differen-tial operators on G. As in the previous section these equations can beanalyzed in detail, and it can be shown that the integrals behave asymp-totically like

E([P ], ξ, λ, ψ)(k1m expX k2)

∼∑

w∈W (aQ|aP )

e(iwλ−ρQ)(X) [cQ|P,ξ(w, λ)ψ](m, k1, k2)

10 Analysis on groups and symmetric spaces 57

for m ∈ MQ, k1, k2 ∈ K, and as X tends to infinity in a+Q; here Q is a

parabolic subgroup in the same equivalence class as P and W (aQ|aP )denotes the finite set of isomorphisms aP → aQ induced by the ad-joint action of K. Each coefficient cQ|P,ξ(w, λ) is an isomorphism fromthe finite dimensional Hilbert space A2(MP , ξ, δ) onto the similar spaceA2(MQ, wξ, δ). It can be shown that

cQ|P,ξ(w, λ)∗cQ|P,ξ(w, λ) = η(P, ξ, λ) I

with η(P, ξ, λ) a strictly positive scalar, independent of Q,w, δ anddepending real analytically on λ ∈ a∗P . Finally, the measure for thePlancherel decomposition of L2(G)[P ],ξ is given by

dλ

η(P, ξ, λ). (10.1)

In this sense, Weyl’s principle is valid for all continuous spectral param-eters in the Plancherel decomposition for G.

In the 1980’s and 1990’s, much progress was made in harmonic analysison general semisimple symmetric spaces. These are pseudo-Riemanniansymmetric spaces of the form G/H, with G a real semisimple Lie groupand H (an open subgroup of) the group of fixed points for an involutionσ of G. This class of spaces contains both the Riemannian symmetricspaces and the semisimple groups. Indeed the group G is a homogeneousspace for the action of G×G given by (x, y) · g = xgy−1. The stabilizerof the identity element eG equals the diagonal H of G×G, which is thegroup of fixed points for the involution σ : (x, y) 7→ (y, x). As a decom-position for the left times right regular action of G × G on L2(G) thePlancherel decomposition becomes multiplicity free. This is analogousto what happens for the Peter-Weyl decomposition for compact groups.

Another interesting class of semisimple symmetric spaces is formed bythe pseudo-Riemannian hyperbolic spaces SOe(p, q)/SOe(p−1, q), p > 1.

For general semisimple symmetric spaces, M. Flensted-Jensen, [9],gave the first construction of discrete series assuming the analogue ofHarish-Chandra’s rank condition. The full classification of the discreteseries was then given by T. Oshima and T. Matsuki [25].

In [2], E.P. van den Ban and H. Schlichtkrull gave a description ofthe most continuous part of the Plancherel decomposition. Here, a newphenomenon is that the Plancherel decomposition may have finite mul-tiplicities. Nevertheless, the multiplicities can be parametrized in such away that Weyl’s principle generalizes to this context. Then, P. Delorme,partly in collaboration with J. Carmona, determined the full Plancherel


decomposition for G/H, [6], [8]. Around the same time this was alsoachieved by E.P. van den Ban and H. Schlichtkrull, [3],[4], with a com-pletely different proof. In all these works, the appropriate analogue of(10.1) goes through. For more information, we refer the reader to thesurvey articles in [1].

Parallel to the developments sketched above, G. Heckman and E. Op-dam [19] developed a theory of hypergeometric functions, generalizingthe elementary spherical functions of the Riemannian symmetric spaces.For these spaces, the algebra of radial components of invariant differen-tial operators is entirely determined by a root system and root multi-plicities. The generalization is obtained by allowing these multiplicitiesto vary in a continuous fashion. In the associated Plancherel decompo-sition, established by Opdam, [24], Weyl’s principle holds through theanalogue of (9.6).

11 Appendix: circles in P1(C)

If V is a two dimensional complex linear space, then by P(V ) we denotethe 1-dimensional projective space of lines Cv, with v ∈ V \ 0. In anatural way we will identify subsets of P(V ) with C-homogeneous subsetsof V containing 0. In particular, the empty set is identified with 0. Thegroup GL(V ) of invertible complex linear transformations of V naturallyacts on P(V ).

Let β be Hermitian form on V, i.e., β : V ×V → C is linear in the firstand conjugate linear in the second component, and β(v, w) = β(w, v)for all v, w ∈ V. By symmetry, β(v, v) ∈ R for all v ∈ V. We denote by Bthe space of Hermitian forms β on V for which the function v 7→ β(v, v)has image R. Equivalently, this means that there exists a basis v1, v2 ofV such that β(v1, v1) = 1 and β(v2, v2) = −1. It follows from this thatthe group GL(V ) acts transitively on B by g · β(v, w) = β(g−1v, g−1w).

We note that for any Hermitian form β on V the map v 7→ β(v, · )induces a linear map from V to the conjugate linear dual space V

∗. This

map is an isomorphism if and only if β is non-degenerate. Let γ be anychoice of positive definite Hermitian form on V. Then Hβ = γ−1 β isa linear endomorphism of V ; from β(v, w) = γ(Hβv, w) for v, w ∈ V

we see that Hβ is symmetric with respect to the inner product γ. Thecondition that β ∈ B is equivalent to the condition that Hβ has botha strictly positive and a strictly negative eigenvalue, which in turn isequivalent to the condition that detHβ < 0. For obvious reasons we willcall B the space of Hermitian forms of signature (1, 1).

11 Appendix: circles in P1(C) 59

By a circle in P(V ) we mean a set of the form

Cβ := v ∈ V | β(v, v) = 0

with β ∈ B. For g ∈ GL(V ) we have g(Cβ) = Cg·β so that the naturalaction of GL(V ) on the collection of circles is transitive.

We now turn to the case of C2 equipped with the standard Hermitianinner product. Accordingly, any form β ∈ B is represented by a uniqueHermitian matrix H of strictly negative determinant. We will use thestandard embedding C → P1(C) := P(C2) given by z 7→ C(z, 1). Thecomplement of the image of this embedding consists of the single point∞C := C(1, 0). The inverse map χ : P1(C) \ ∞C → C is called thestandard affine chart. It is straightforwardly verified that ∞C belongs toCβ if and only if the entry H11 equals zero. In this case the intersectionof Cβ with the standard affine chart is given by 2Re (H21z) = −H22,

which is the straight line −H−121 ( 1

2H22 + iR). In particular, the form

i[z, w] = i(z1w2 − z2w1) (11.1)

is represented by the Hermitian matrix iJ (see (2.4)), and the associatedcircle in P1(C) equals the closure P1(R) := CR2 = R ∪ ∞C of the realline.

In the remaining case the circle Cβ is completely contained in thestandard affine chart, and in the affine coordinate it equals a circle withrespect to the standard Euclidean metric on C ' R2. The radius r andthe center α are given by

r2 = − detH|H11|2

, α = −H12

H11. (11.2)

The preimage under χ of the interior of this circle is the subset of P1(C)given by the inequality

sign(H11)β(z, z) < 0.

We note that all circles and straight lines in C are representable in theabove fashion. In the standard affine coordinate, the action of the groupGL(2,C) on P1(C) is represented by the action through fractional lineartransformations on C. Accordingly, we retrieve the well-known fact thatthis action preserves the set of circles and straight lines.

More generally, let v1, v2 be a complex basis of V. Then the naturalmap z 7→ z1v1 + z2v2 induces a diffeomorphism v : P1(C) → P(V ).The map χv := χ v−1 : P(V ) \ Cv1 → C is said to be the affine chartdetermined by v1, v2. Note that z = χv(C(zv1 + v2)), for z ∈ C. The


general linear group GL(V ) acts on the set of affine charts by (g, ψ) 7→ψ g−1, so that g · χv = χgv. Clearly, the action is transitive. It followsthat the transition map between any pair of affine charts is given by afractional linear transformation.

From the above considerations it follows that a circle C in P(V ) cor-responds to a circle in the affine chart χv if and only if Cv1 does not lieon C. Otherwise, the circle is represented by a straight line in χv.

References

[1] J.-P. Anker and B. Orsted, editors. Lie theory, volume 230 of Progressin Mathematics. Birkhauser Boston Inc., Boston, MA, 2005. Harmonicanalysis on symmetric spaces—general Plancherel theorems.

[2] E. P. van den Ban and H. Schlichtkrull. The most continuous part of thePlancherel decomposition for a reductive symmetric space. Ann. of Math.(2), 145:267–364, 1997.

[3] E. P. van den Ban and H. Schlichtkrull. The Plancherel decompositionfor a reductive symmetric space. I. Spherical functions. Invent. Math.,161:453–566, 2005.

[4] E. P. van den Ban and H. Schlichtkrull. The Plancherel decomposition fora reductive symmetric space. II. Representation theory. Invent. Math.,161:567–628, 2005.

[5] A. Borel. Essays in the history of Lie groups and algebraic groups, vol-ume 21 of History of Mathematics. American Mathematical Society, Prov-idence, RI, 2001.

[6] J. Carmona and P. Delorme. Transformation de Fourier sur l’espace deSchwartz d’un espace symetrique reductif. Invent. Math., 134:59–99, 1998.

[7] E. A. Coddington and N. Levinson. Theory of ordinary differential equa-tions. McGraw-Hill Book Company, Inc., New York-Toronto-London,1955.

[8] P. Delorme. Formule de Plancherel pour les espaces symetriques reductifs.Ann. of Math. (2), 147:417–452, 1998.

[9] M. Flensted-Jensen. Discrete series for semisimple symmetric spaces. Ann.of Math. (2), 111:253–311, 1980.

[10] R. Gangolli. On the Plancherel formula and the Paley-Wiener theoremfor spherical functions on semisimple Lie groups. Ann. of Math. (2),93:150–165, 1971.

[11] S. G. Gindikin and F. I. Karpelevic. Plancherel measure for symmetricRiemannian spaces of non-positive curvature. Dokl. Akad. Nauk SSSR,145:252–255, 1962.

[12] Harish-Chandra. Spherical functions on a semisimple Lie group. I. Amer.J. Math., 80:241–310, 1958.

[13] Harish-Chandra. Spherical functions on a semisimple Lie group. II. Amer.J. Math., 80:553–613, 1958.

[14] Harish-Chandra. Discrete series for semisimple Lie groups. I. Constructionof invariant eigendistributions. Acta Math., 113:241–318, 1965.

61

62 References

[15] Harish-Chandra. Discrete series for semisimple Lie groups. II. Explicitdetermination of the characters. Acta Math., 116:1–111, 1966.

[16] Harish-Chandra. Harmonic analysis on real reductive groups. I. The the-ory of the constant term. J. Functional Analysis, 19:104–204, 1975.

[17] Harish-Chandra. Harmonic analysis on real reductive groups. II.Wavepackets in the Schwartz space. Invent. Math., 36:1–55, 1976.

[18] Harish-Chandra. Harmonic analysis on real reductive groups. III. TheMaass-Selberg relations and the Plancherel formula. Ann. of Math. (2),104:117–201, 1976.

[19] G. J. Heckman and E. M. Opdam. Root systems and hypergeometricfunctions. I. Compositio Math., 64:329–352, 1987.

[20] S. Helgason. An analogue of the Paley-Wiener theorem for the Fouriertransform on certain symmetric spaces. Math. Ann., 165:297–308, 1966.

[21] D. Hilbert. Grundzuge einer allgemeinen Theorie der linearen Integral-gleichungen. Chelsea Publishing Company, New York, N.Y., 1953.

[22] I.S. Kac. The existence of spectral functions of generalized second orderdifferential systems with boundary conditions at the singular end. Amer.Math. Soc. Transl. (2), 62:204–262, 1964.

[23] K. Kodaira. The eigenvalue problem for ordinary differential equations ofthe second order and Heisenberg’s theory of S-matrices. Amer. J. Math.,71:921–945, 1949.

[24] E. M. Opdam. Harmonic analysis for certain representations of gradedHecke algebras. Acta Math., 175:75–121, 1995.

[25] T. Oshima and T. Matsuki. A description of discrete series for semisimplesymmetric spaces. In Group representations and systems of differentialequations (Tokyo, 1982), volume 4 of Adv. Stud. Pure Math., pages 331–390. North-Holland, Amsterdam, 1984.

[26] J. Rosenberg. A quick proof of Harish-Chandra’s Plancherel theorem forspherical functions on a semisimple Lie group. Proc. Amer. Math. Soc.,63:143–149, 1977.

[27] E. C. Titchmarsh. Eigenfunction Expansions Associated with Second-Order Differential Equations. Oxford, at the Clarendon Press, 1946.

[28] H. Weyl. Singulare Integralgleichungen. Math. Ann., 66:273–324, 1908.

[29] H. Weyl. Uber gewohnliche Differentialgleichungen mit Singularitaten unddie zugehorigen Entwicklungen willkurlicher Funktionen. Math. Ann.,68:220–269, 1910.

[30] H. Weyl. Uber gewohnliche lineare Differentialgleichungen mit singularenStellen und ihre Eigenfunktionen. (2. Note). Gottinger Nachrichten, pp.442–467, 1910.

[31] H. Weyl. Ramifications, old and new, of the eigenvalue problem. Bull.Amer. Math. Soc., 56:115–139, 1950.

weyl, eigenfunction expansions and harmonic …symmetric spaces of rank 1. it is the purpose of this...

Documents